Reflectionless measures for Calderón-Zygmund operators

Abstract

We study the properties of reflectionless measures for a Calderón-Zygmund operator T. Roughly speaking, these are measures $\mu$ for which T(μ) vanishes (in a weak sense) on the support of the measure. We describe the relationship between certain well-known problems in harmonic analysis and geometric measure theory and the classification of reflectionless measures. As an application of our theory, we give a new proof of a recent theorem of Eiderman, Nazarov, and Volberg, which states that in ${\mathbb{R}}^d$, the s-dimensional Riesz transform of a non-trivial $s$-dimensional measure is unbounded if $s\in (d-1,d)$.

34 pages. This is a preliminary version: a minor revision of the paper will take place over the next fortnight or so

References in corpus (2)

• On the uniform rectifiability of AD regular measures with bounded Riesz transform operator: the case of codimension 1
• Calderon-Zygmund capacities and Wolff potentials on Cantor sets